Introduction to Nonequilibrium Statistical Mechanics

[Count Iblis]

http://ptp.ipap.jp/link?PTP/123/581/"

In this article, we present a concise and self-contained introduction to nonequilibrium statistical mechanics with quantum field theory by considering an ensemble of interacting identical bosons or fermions as an example. Readers are assumed to be familiar with the Matsubara formalism of equilibrium statistical mechanics such as Feynman diagrams, the proper self-energy, and Dyson's equation. The aims are threefold: (i) to explain the fundamentals of nonequilibrium quantum field theory as simple as possible on the basis of the knowledge of the equilibrium counterpart; (ii) to elucidate the hierarchy in describing nonequilibrium systems from Dyson's equation on the Keldysh contour to the Navier-Stokes equation in fluid mechanics via quantum transport equations and the Boltzmann equation; (iii) to derive an expression of nonequilibrium entropy that evolves with time. In stage (i), we introduce nonequilibrium Green's function and the self-energy uniquely on the round-trip Keldysh contour, thereby avoiding possible confusions that may arise from defining multiple Green's functions at the very beginning. We try to present the Feynman rules for the perturbation expansion as simple as possible. In particular, we focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, i.e., Baym's Φ-derivable approximation, which has a crucial property for nonequilibrium systems of obeying various conservation laws automatically. We also show how the two-particle correlations can be calculated within the Φ-derivable approximation, i.e., an issue of how to handle the “Bogoliubov-Born-Green-Kirkwood-Yvons (BBGKY) hierarchy”. Aim (ii) is performed through successive reductions of relevant variables with the Wigner transformation, the gradient expansion based on the Groenewold-Moyal product, and Enskog's expansion from local equilibrium. This part may be helpful for convincing readers that nonequilibrium systems can be handled microscopically with quantum field theory, including fluctuations. We also discuss a derivation of the quantum transport equations for electrons in electromagnetic fields based on the gauge-invariant Wigner transformation so that the Lorentz force is reproduced naturally. As for (iii), the Gibbs entropy of equilibrium statistical mechanics suffers from the flaw that it does not evolve in time. We show here that a microscopic expression of nonequilibrium dynamical entropy can be derived from the quantum transport equations so as to be compatible with the law of increase in entropy as well as equilibrium statistical mechanics.

 

[eljose79]

Dear forum members,

I am glad to present to you this article on nonequilibrium statistical mechanics with quantum field theory. This is a very important and interesting topic in the field of physics, as it allows us to understand the behavior of systems that are far from equilibrium, such as in the case of chemical reactions, fluid dynamics, and many other phenomena.

The article aims to provide a clear and concise introduction to this subject, assuming that the readers are already familiar with the Matsubara formalism of equilibrium statistical mechanics. The authors have three main goals: (i) to explain the fundamentals of nonequilibrium quantum field theory in a simple manner, building upon the knowledge of the equilibrium counterpart, (ii) to show the hierarchy of describing nonequilibrium systems from Dyson's equation to the Navier-Stokes equation, and (iii) to derive an expression for nonequilibrium entropy that evolves with time.

To achieve these goals, the article introduces the concept of nonequilibrium Green's function and the self-energy on the round-trip Keldysh contour, which helps avoid any confusion that may arise from defining multiple Green's functions. The authors also present the Feynman rules for perturbation expansion in a simple manner. They focus on the self-consistent perturbation expansion with the Luttinger-Ward thermodynamic functional, which has the important property of automatically obeying various conservation laws for nonequilibrium systems.

The article also discusses the reduction of relevant variables with the Wigner transformation, gradient expansion, and Enskog's expansion, which shows that nonequilibrium systems can be handled microscopically with quantum field theory, taking into account fluctuations. The authors also derive the quantum transport equations for electrons in electromagnetic fields, based on the gauge-invariant Wigner transformation.

Lastly, the article addresses the issue of nonequilibrium entropy and its evolution in time. While the Gibbs entropy of equilibrium statistical mechanics does not evolve in time, the authors show how a microscopic expression for nonequilibrium dynamical entropy can be derived from the quantum transport equations, which is in accordance with the law of increase in entropy.

In conclusion, this article provides a comprehensive and clear explanation of nonequilibrium statistical mechanics with quantum field theory. It is a valuable resource for anyone interested in this field and will greatly contribute to our understanding of nonequilibrium systems. Thank you for reading and I hope you find this article informative and useful.